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A generalized Brans-Dicke (GBD) theory in the framework of Palatini formalismare proposed in this paper. We derive the field equations by using thevariational approach and obtain the linearized equations by using theweak-field approximation method. We show various properties of the geometricalscalar field in the Palatini-formalism of GBD theory: it is massless andsource-free, which are different from the results given in the metric-formalismof GBD theory. Also, we investigate the polarization modes of gravitationalwaves (GWs) by using the the geodesic deviation method and the Newman-Penrosemethod in the Palatini-GBD theory. It is observed that there are threepolarizations modes and four oscillation in the Palatini-GBD theory.Concretely, they are the two transverse tensor ($+$) and ($\times$) standardpolarization modes, and one breathing mode (with two oscillation). The resultsof GWs polarization in the Palatini-GBD theory are different from that in themetric-GBD theory, where there are four polarizations modes: the two standardtensorial modes ($+$ and $\times$), a scalar breathing mode, and a massivescalar mode that is a mix of the longitudinal and the breathing polarization.Comparing with the Palatini-$f(\tilde{R})$ theory and the General Relativity,we can see that the extra breathing mode of GWs polarization can be found inthe Palatini-GBD theory. At last, the expression of the parameterized postNewton (PPN) parameter is derived, which could pass through the experimentaltest.

Linearized physics and gravitational-waves polarizations in the Palatini formalism of GBD theory